Vector and Geometric Calculus (Geometric Algebra & Calculus) 1480132454, 9781480132450
This textbook for the undergraduate vector calculus course presents a unified treatment of vector and geometric calculus
348 20 7MB
English Pages 211 [186] Year 2012
Table of contents :
Contents
Preface
I Preliminaries
1 Curve and Surface Representations
1.1 Curve Representations
1.2 Surface Representations
1.3 Polar, Cylindrical, Spherical Coordinates
2 Limits and Continuity
2.1 Open and Closed Sets
2.2 Limits
2.3 Continuity
II Derivatives
3 The Differential
3.1 The Partial Derivative
3.2 The Taylor Expansion
3.3 The Differential
3.4 The Chain Rule
3.5 The Directional Derivative
3.6 Inverse and Implicit Functions
4 Tangent Spaces
4.1 Manifolds
4.2 Tangent Spaces to Curves
4.3 Tangent Spaces to Surfaces
5 The Gradient
5.1 Fields
5.2 The Gradient
5.3 Scalar and Vector Fields
5.4 Curvilinear Coordinates
5.5 The Vector Derivative
6 Extrema
6.1 Extrema
6.2 Constrained Extrema
III Integrals
7 Integrals over Curves
7.1 The Scalar Integral
7.2 The Path Integ ral
7.3 The Line Integral
7.4 Conservative Vector Fields
8 Multiple Integrals
8.1 Multiple Integrals
8.2 Change of Variables
9 Integrals over Surfaces
9.1 The Surface Integral
9.2 The Flux Integral
IV The Fundamental Theorem of Calculus
10.1 The Fundamental Theorem of Calculus
10.2 The Divergence Theorem
10.3 The Curl Theorem
10.4 Analytic Functions
V Differential Geometry
11 Differential Geometry in R³
11.1 Curves
11.2 Surfaces
11.3 Curves in Surfaces
VI Appendices
A Review of Geometric Algebra
B Software
C Formulas
D Differential Forms
Index
Contents
Preface
I Preliminaries
1 Curve and Surface Representations
1.1 Curve Representations
1.2 Surface Representations
1.3 Polar, Cylindrical, Spherical Coordinates
2 Limits and Continuity
2.1 Open and Closed Sets
2.2 Limits
2.3 Continuity
II Derivatives
3 The Differential
3.1 The Partial Derivative
3.2 The Taylor Expansion
3.3 The Differential
3.4 The Chain Rule
3.5 The Directional Derivative
3.6 Inverse and Implicit Functions
4 Tangent Spaces
4.1 Manifolds
4.2 Tangent Spaces to Curves
4.3 Tangent Spaces to Surfaces
5 The Gradient
5.1 Fields
5.2 The Gradient
5.3 Scalar and Vector Fields
5.4 Curvilinear Coordinates
5.5 The Vector Derivative
6 Extrema
6.1 Extrema
6.2 Constrained Extrema
III Integrals
7 Integrals over Curves
7.1 The Scalar Integral
7.2 The Path Integ ral
7.3 The Line Integral
7.4 Conservative Vector Fields
8 Multiple Integrals
8.1 Multiple Integrals
8.2 Change of Variables
9 Integrals over Surfaces
9.1 The Surface Integral
9.2 The Flux Integral
IV The Fundamental Theorem of Calculus
10.1 The Fundamental Theorem of Calculus
10.2 The Divergence Theorem
10.3 The Curl Theorem
10.4 Analytic Functions
V Differential Geometry
11 Differential Geometry in R³
11.1 Curves
11.2 Surfaces
11.3 Curves in Surfaces
VI Appendices
A Review of Geometric Algebra
B Software
C Formulas
D Differential Forms
Index
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V ector and G eom etric Calculus A la n M a c d o n a ld Luther College, Decorah, IA 52101 USA [email protected] faculty.luther.edu / "macdonal
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Geometry without algebra is dumb! - Algebra without geometry is blind! - David Hestenes
T he principal argument for the adoption of geometric algebra is th at it provides a single, simple mathematical framework which elimi nates the plethora of diverse m athematical descriptions and tech niques it would otherwise be necessary to learn. - Allan McRobie and Joan Lasenbv
To. Ellen
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